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IN  MEMORIAM 
FLORIAN  CAJORI 


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PRINCIPLES 


OF 


PLANE    GEOMETRY 


BY 


J.  W.  MACDONALD 

PRINCIPAL    OF   THE   STONEHAM    (MASS.)    HIGH    SCHOOL 


Let  him  know  a  thing  because  he  has  fatind  it  out  for  himself,  and  not 
because  you  have  told  him.  —  J.  J.  Rousseau 


33o6ton 

ALLYN      AND      BACON 

1894 


<}»'»■ 


PRINCIPLES 


OF 


PLANE    GEOMETRY 


BY 

J.  w.Jmacdonald 

PRINCIPAL   OF   THE   STONEHAM    (MASS.)    HIGH    SCHOOL 


Let  him  know  a  thing  because  he  has  found  it  out  for  himself,  and  not 
because  you  have  told  him.  —  J.  J.  Rousseau 


93aston 

allVn    and    bacon 
1894 


CAJORI 

Copyright,  1889, 
By  J.  W.  MacDonald. 


BSttibcrstta  Ipregs: 
John  Wilson  and  Son,  Cambridge. 


PREFACE. 


The  most  appropriate  preface  for  a  book  of  this 
kind  would  be  an  exposition  of  the  principles  of 
psychology  pertaining  to  the  development  and  train- 
ing of  the  reasoning  and  linguistic  faculties.  As 
such  a  preface,  however,  would  be  more  pretentious 
than  the  book  proper,  and  as  these  principles  have 
been  so  well  expounded  by  many  eminent  writers,  I 
must  content  myself  by  merely  urging  studious  in- 
vestigation upon  all  teachers  who  are  ambitious  to 
practise  the  best  methods.  If  teachers  will  do  this, 
of  one  thing  I  am  confident,  —  they  will  grant  that 
the  purpose  of  this  book  is  right  in  theory,  even  if  in 
practice  certain  difficulties  may  seem  to  them  insur- 
mountable. As  a  help,  however,  to  teachers  who 
may  wish  it,  I  have  thought  it  advisable  to  publish 
as  a  companion  to  this  book  a  monograph  on  teach- 
ing geometry,*  illustrating  actual  class-work,  and 
showing  in  detail  how  some  of  the  most  difficult 
topics    may  be   mastered,  —  not,   let   me   add,  as   a 

*  Geometry  in  the  Secondary  School.  Willard  Small,  Publisher, 
24  Franklin  Street,  Boston,  Mass. 

/A 


iv  PREFACE. 


dogmatic  declaration  of  tJie  method  of  development, 
but  only  as  a  suggestive  illustration  of  a  method. 
The  earnest  teacher,  who  thoroughly  understands 
the  subject  he  is  teaching,  and  who  has  a  purpose 
clear  in  his  mind,  will  make  his  own,  and  for  him  the 
best  method. 

It  cannot  be  said  of  this  Geometry,  as  I  have  heard 
it  said  of  others,  that  it  is  designed  to  aid  inefficient 
teachers.  The  teacher  who  does  not  thoroughly  un- 
derstand elementary  geometry,  —  who  is  not  sharp  to 
detect  inaccuracies  in  definitions  and  arguments, — 
who,  in  short,  is  dependent  on  the  written  text  for 
what  he  teaches,  —  should  not  undertake  to  use  this 
book. 

I  have  thought  it  best  to  publish  the  books  in  an 
inexpensive  form,  so  that,  where  the  free  text-book 
system  exists,  it  would  be  as  economical  to  purchase 
them  new  each  year  as  to  transmit  a  more  expen- 
sive volume  from  pupil  to  pupil,  with  much  distasteful 
accumulation.  They  will  be  furnished  in  a  more 
durable  form,   if  desired. 

In  conclusion,  I  wish  to  thank  the  publishers  and 
the  proof-reader,  by  whose  suggestions  and  watch- 
fulness the  text  has  been  much  improved  and  saved 
from  numerous  errors. 

J.  W.  M. 

Stoneham,  August  I,  1889. 


CONTENTS. 


Fagb 

Definitions  of  Solids,  Surfaces,  etc i 

Locus  OF  A  Point 4 

Position  of  Lines 4 

Plane  Angles 4 

Axioms 5 

Postulates 5 

Propositions 6 

Symbols  and  Abbreviations 6 

BOOK   L 

Angles  having  Special  Names 9 

Triangles 12 

Quadrilaterals 16 

Polygons  of  more  than  Four  Sides 18 

Axis  of  Symmetry,  etc 18 

Supplementary  Propositions 20 

BOOK  n. 

Ratio 22 

Proportions 22 

The  Theory  of  Limits 25 


Vi  CONTENTS. 


BOOK    III. 

Pace 

The  Circle 29 

Inscribed  Angles  and  Polygons 30 

Supplementary  Propositions 38 

BOOK   IV. 

Similar  Polygons 43 

Dividing  Lines  Internally  and  Externally  ...  45 

Dividing  Lines  Harmonically     . 45 

Extreme  and  Mean  Ratio 47 

Supplementary  Propositions 48 

BOOK  V. 

Measurement  of  Areas 50 

Projection 55 

Special  Problems  in  Triangles 56 

BOOK  VI. 

Regular  Polygons  58 

Special  Problems  in  Inscribed  Polygons  ....  63 


PLANE     GEOMETRY, 


DEFINITIONS. 
I.  Solids. — Surfaces. — Lines. — Points. 

1.  What  is  a  geometrical  solid  ? 

a.  What  dimensions  has  it  ? 

b.  How  is  it  bounded? 

2.  What  is  a  surface  ?     Its  dimensions  ? 

a.  A  plane  surface  ? 

b.  A  curved  surface  ? 

c.  How  is  a  surface  bounded? 

3.  What  is  a  line  ?     Its  dimension  ? 

a.  What  is  a  straight  line  or  right  line  ? 

b.  What  is  a  curved  line  ? 

c.  What  is  a  broken  line  ? 

Illustration. 

d.  What  is  a  mixed  line  ? 
Illustration. 

Note.    The  mixed  line  and  the  broken  line  have  no  practical  value 
in  geometrical  discussions. 

I 


PLANE  GEOMETRY. 


4.  What  is  a  point  ? 

5.  How  may  a  line  be  generated  ? 

6.  How  may  a  surface  be  generated? 

7.  How  may  a  solid  be  generated  ? 

8.  Wliat  determines  the  position  of  a  point  ? 

9.  What  determines  the  position  of  a  line  ? 

10.  What  determines  the  position  of  a  surface  ? 

11.  What  determines  the  form  of  a  surface  ? 

12.  What  determines  the  form  of  a  soUd? 


13.  What  is  a  figure  ? 

a.  A  plane  figure  ? 

Illustrations. 


o 


b.   A  rectilinear  figure  ? 

Illustrations, 


c.   A  curvilinear  figure  ? 

Illustrations. 


PLANE   GEOMETRY. 


d.   A  mixtilinear  figure  ? 

Illustrations 


e.   Similar  figures  ? 


Illustrations. 


(^  ^)  (O 


/.   Equivalent  figures  ? 


Illustrations. 


g.    Equal  figures? 


Illustrations. 


'/4.5q.\\ '/45Q. 
INchW  inch 


h.   Solid  figures  ? 


Illustrations. 


14.   What  is  magnitude  ? 

a.  Of  a  line  ? 

b.  Of  a  surface  ? 

c.  Of  a  solid  ? 


r\ 


PLANE   GEOMETRY. 


15.  How  is  magnitude  measured  ? 

a.  Of  a  line  ? 

b.  Of  a  surface  ? 
^.   Of  a  solid  ? 

16.  What  is  geometry  ? 

a.  Solid  geometry? 

b.  Plane  geometry? 

LOCUS. 

17.  What  is  the  locus  of  a  point  ? 

18.  Problem  I.     Find  the  locus  of  a  point  a  given  dis- 
tance from  a  given  point. 

Problem  II.     Find  the  locus  of  a  point  a  given  distance 
from  a  given  circumference. 


II.  Positions  of  Lines.  —  Plane  Angles. 

19.  What  are  parallel  lines  ? 

20.  What  is  a  perpendicular  line  ? 

21.  What  is  an  oblique  line  ? 

22.  What  is  an  angle  ? 

a.   A  right  angle  ? 

Illustration. 


b.   An  acute  angle  ? 
Illustration. 


PLANE   GEOMETRY. 


.   An  obtuse  angle  ? 

Illustration.  ^\!^'^. 


C 


d.  A  straight  angle  ? 

Illustration.     B — l ^ C 

e.  A  reflex  angle  ? 

Illustration. 


23.  What  are  complementary  angles  ? 

24.  What  are  supplementary  angles  ? 

III.  Axioms. — Postulates.  —  Propositions. 

25.  What  is  an  axiom  ? 

26.  What  axioms  can  be  formed  from  the  following  data? 

a.  Things  equal  to  the  same  thing. 

b.  Adding  equals. 

c.  Adding  unequals. 

d.  Subtracting  equals. 

e.  Subtracting  unequals. 
/.  Multiplying  equals. 
g.  Dividing  equals. 

h.  The  whole  and  a  part. 
/.   The  whole  and  all  the  parts. 

27.  What  axioms  may  be  asserted  as  to  the  equality  of 

a.  Right  angles  ? 

b.  Straight  angles  ? 

c.  Complementary  angles? 

d.  Supplementary  angles?  _ 

28.  What  is  a  postulate  ? 


PLANE   GEOMETRY. 


29.  What  must  be  granted  as  to  the  following : 

a.  Drawing  a  straight  Hne  ? 

b.  Prolonging  a  line  ? 

c.  Drawing  a  circumference  ? 

d.  Dividing  lines,  angles,  etc.  ? 

30.  What  is  a  proposition? 

a.  A  problem  ? 

b.  A  theorem  ? 

c.  A  corollary? 

d.  A  scholium? 

31.  Of  what  does  every  theorem  consist? 

IV.  Mathematical  Symbols  and  Abbreviations. 

32.  a.  + ,  plus. 

b.  —,  minus. 

c.  X ,  multiplied  by. 

d.  -^,  divided  by. 

e.  =,  equal  to. 

/.  =s=,  equivalent  to. 

g.  >,  greater  than. 

^.  < ,  less  than. 

/.  :,   ::,   :,  signs  of  proportion. 

/  Z ,  angle ;  A ,  angles. 

^.  A ,  triangle ;  A ,  triangles. 

/.  n,  square;  [s],  squares. 

m.  O,  parallelogram  ;  lU,  parallelograms. 

n.  O ,  circle  ;  © ,  circles. 

0. 


BOOK    I. 


Proposition  I.     A  Theorem. 

33.  If  one  straight  line  meets  another  so  as  to  form 
two  adjacent  angles,  the  sura  of  these  angles  is  equal  to 
two  right  angles ;  that  is,  the  angles  are  supplements  of 
each  other. 

Corollary  I.  Any  number  of  angles  in  the  same  plane, 
formed  about  a  given  point  on  one  side  of  a  straight  line, 
are  equivalent  to  two  right  angles. 

Corollary  II.  The  sum  of  all  the  angles  in  the  same 
plane,  formed  about  a  given  point,  is  equal  to  four  right 
angles. 

Proposition  II.    A  Theorem. 

34.  Conversely,  if  two  angles  whose  sum  equals  two  right 
angles  are  placed  adjacent  to  each  other,  their  exterior  sides 
will  form  one  straight  line. 


Proposition  III.    A  Theorem. 

35.    If  two  straight  lines  intersect  each  other,  the  vertical 
angles  are  equal. 

See  page  5,  §§  24  and  27  d. 


PLANE  GEOMETRY. 


Proposition  IV.    A  Theorem. 

36.  At  any  point  in  a  straight  line  there  can  be  but  one 
perpendicular  on  either  side. 

Proposition  V.    A  Theorem. 

37.  From  any  point  outside  of  a  straight  line  there  can 
be  but  one  perpendicular  to  the  line. 

Proposition  VI.    A  Theorem. 

38.  Two  lines  in  the  same  plane  perpendicular  to  a  third 
line  are  parallel  to  each  other. 

Proposition  VII.    A  Theorem. 

39.  If  a   straight   line  is   perpendicular  to  one  of  two 
parallel  lines,  it  is  perpendicular  to  the  other  also. 

Proposition  VIII.    A  Theorem. 

40.  Angles  having  the  sides  of  the  one  parallel  to  the  sides 
of  the  other  are  either  equals  or  supplements. 

Scholium.  If  both  pairs  of  parallel  sides  extend  in  the 
same  direction  from  the  vertices,  or  both  in  opposite  direc- 
tions, the  angles  are  equal ;  but  if  one  pair  extends  in  the 
same  direction  and  the  other  pair  in  opposite  directions, 
the  angles  are  supplements. 

Proposition  IX.    A  Theorem. 

41.  Angles  having  the  sides  of  the  one  perpendicular  to 
the  sides  of  the  other  are  either  equals  or  supplements. 

Scholium.  How  can  it  be  determined  whether  the  angles 
are  equals  or  supplements? 


BOOK   I. 


V.    Angles  having  Special  Names. 

42.  Let  two  straight  lines  be  intersected  by  a  third  : 

a.  What  are  the  exterior  angles  ? 

b.  What  are  the  interior  angles  ? 

c.  What  are  alternate  exterior  angles  ? 

d.  What  are  alternate  interior  angles  ? 

e.  What  are  external  angles  on  the  same  side  (of 

the  intersecting  line)  ? 
/.   What  are  internal  angles  on  the  same  side  ? 
g.   What  are  opposite  external-internal  angles  ? 

Proposition  X.    A  Theorem. 

43.  If  two   parallel   straight  lines   are   intersected   by  a 
third  : 

I.    Alternate  interior  or  exterior  angles  will  be  equal. 

II.    Opposite  external-internal  angles  will  be  equal. 
III.    Interior  or  exterior  angles  on  the  same  side  will 
be  supplements. 

See  Proposition  VIII. 

Proposition  XI.    A  Tiieorem. 

44.  Two  straight  lines  intersected  by  a  third  line  will  be 
parallel : 

I.   If  alternate  interior  or  exterior  angles  are  equal. 
II.    If  opposite  external-internal  angles  are  equal. 
III.    If  interior  or  exterior  angles  on  the  same  side  are 
supplements. 


PLANE   GEOMETRY. 


Proposition  XII.    A  Theorem. 

45.  Two  lines  parallel  to  a  third  are  parallel  to  each  other. 
See  Proposition  VII . 

Proposition  XIII.    A  Theorem. 

46.  The  sum  of  two  lines  drawn  from  any  point  to  the  ex- 
tremities of  a  line  is  greater  than  the  sum  of  any  two  lines 
similarly  drawn  from  an  included  point. 

Proposition  XIV.    A  Theorem. 

47.  The  shortest  distance  from  any  point  to  a  given 
straight   line  is  a  perpendicular  to  that  line. 

Proposition  XV.    A  Theorem. 

48.  Two  oblique  lines  extending  from  any  point  in  a  per- 
pendicular to  points  in  the  base  line  equally  distant  from  the 
foot  of  the  perpendicular  are  equal. 

Proposition  XVI.    A  Theorem. 

49.  Of  two  oblique  lines  extending  from  any  point  in  a 
perpendicular  to  points  in  the  base  line  unequally  distant 
from  the  foot  of  the  perpendicular,  the  one  extending  to  the 
farther  point  will  be  the  longer. 

Corollary.  There  can  be  but  two  equal  oblique  lines 
drawn  from  any  point  in  a  perpendicular  to  the  base  line. 

Proposition  XVII.    A  Theorem. 

50.  If  two  oblique  lines  drawn  from  any  point  in  a  per- 
pendicular to  the  base  line  are  equal,  they  extend  to  points 
equally  distant  from  the  foot  of  the  perpendicular. 


BOOK  I.  II 


Proposition  XVIII.    A  Theorem. 

51.  If  a  perpendicular  be  erected  at  the  middle  point  of  a 
straight  line  : 

I.  Any  point  in  the  perpendicular  will  be  equally  distant 
from  the  extremities  of  the  line. 

II.  Any  point  out  of  the  perpendicular  will  be  unequally 
distant  from  the  extremities  of  the  line. 

Corollary  I.  Conversely,  all  points  equally  distant  from 
I  he  extremities  of  a  line  are  in  the  perpendicular  at  its  middle 
point. 

Corollary  1 1.  The  perpendicular  at  the  middle  point  of 
a  line  will  cut  the  longer  of  two  lines  joining  a  point  with  its 
extremities. 

Proposition  XIX.    A  Problem. 

52.  To  erect  a  perpendicular  at  the  middle  of  a  line. 
6'^<f  page  4,  §§  17  and  i8. 

Proposition  XX.    A  Problem. 

53.  To  bisect  a  given  line. 

Proposition  XXI.    A  Problem. 

54.  To  erect  a  perpendicular  at  any  point  in  a  straight 
Una. 

Proposition  XXII.    A  Problem. 

55.  From  a  point  outside  of  a  straight  line  to  draw  a  per- 
pendicular to  the  line. 


12  PLANE   GEOMETRY 


Proposition  XXIII.    A  Theorem. 

56.  If  an  angle  be  bisected  by  a  straight  line,  every  point 
in  the  bisector  is  equally  distant  from  the  sides. 

Proposition  XXIV.    A  Theorem. 

57.  Conversely,  every  included  point  equally  distant  from 
the  sides  of  an  angle  is  in  the  bisector  of  the  angle. 

Scholium.     What  is  the  locus  of  a  point  equally  distant 
from  the  sides  of  an  angle  ? 


VI.  Triangles. 

58.    What  is  a  triangle  ? 

a.   An  equilateral  triangle  ? 
b*   An  isosceles  triangle  ? 


Illustrations. 


5 

c.  A  scalene  triangle? 

g 
Illustrations.     . 

8 

d.  A  right-angled,  or  right  triangle  ? 

e.  An  obtuse-angled,  or  obtuse  triangle  ? 

f.  An  equiangular  triangle  ? 

59.   What  is  the  hypotenuse  of  a  right  triangle  ? 
a.    What  are  the  other  sides  called  ? 


BOOK   I.  13 


60.  When  are  triangles  equal?     Equivalent? 

a.    What  are  homologous  sides  and  angles  of  equal 
triangles  ? 

61.  What  is  the  base  of  a  triangle  ? 

a.  What  the  altitude  ? 

b.  What  are  its  medians  ? 

Proposition  XXV.    A  Theorem. 

62.  The  sum  of  two  sides  of  a  triangle  is  greater  than  the 
third,  and  the  difference  is  less. 

Proposition  XXVI.    A  Theorem. 

63.  If  two  triangles  have  two  sides  and  the  included  angle 
of  one  equal  to  two  sides  and  the  included  angle  of  the  other, 
each  to  each,  the  other  homologous  parts  are  also  equal,  and 
the  triangles  are  equal. 

Proposition  XXVII.    A  Theorem. 

64.  If  two  triangles  have  two  angles  and  the  included  side 
of  one  equal  to  two  angles  and  the  included  side  of  the 
other,  each  to  each,  the  other  homologous  parts  are  equal, 
and  the  triangles  are  equal. 

Proposition  XXVIII.    A  Theorem. 

65.  If  two  triangles  have  the  three  sides  of  one  equal  to 
the  three  sides  of  the  other,  each  to  each,  the  triangles  are 
equal. 

Proposition  XXIX.    A  Problem. 

66.  Construct  an  equilateral  triangle  having  sides  equal 
each  to  a  given  line. 


14  PLANE   GEOMETRY. 

Proposition  XXX.    A  Problem. 

67.  Construct  a  triangle  having  sides  equal  to  the  sides  of 
a  given  triangle. 

Proposition  XXXI.    A  Theorem. 

68.  If  two  triangles  have  two  sides  of  the  one  respectively 
equal  to  two  sides  of  the  other  and  the  included  angles  un- 
equal, the  third  side  of  the  one  having  the  greater  angle  will 
be  longer  than  the  third  side  of  the  other. 

Scholium.    Three  different  cases  may  arise  ;  prove  each. 

Proposition  XXXII.    A  Theorem. 

69.  Conversely,  if  two  triangles  have  two  sides  of  the  one 
respectively  equal  to  two  sides  of  the  other  and  the  third 
sides  unequal,  the  angle  opposite  the  longer  third  side  will 
be  greater  than  the  angle  opposite  the  shorter. 

Proposition  XXXIII.    A  Theorem. 

70.  The  sum  of  the  angles  of  a  triangle  is  equal  to  two 
right  angles. 

Corollary.  In  a  right  triangle  the  two  acute  angles  are 
complements  of  each  other. 

Scholium.  The  exterior  angle  formed  by  prolonging  one 
of  the  sides  of  a  triangle  is  equal  to  the  sum  of  the  two 
opposite  interior  angles. 

Proposition  XXXIV.     A  Theorem. 

71.  In  an  isosceles  triangle  the  angles  opposite  the  equal 
sides  are  equal. 

Corollary.     An  equilateral  triangle  is  also  equiangular. 


BOOK   I.  15 


Proposition  XXXV.     A  Theorem. 

72.  If  two  angles  of  a  triangle  are  equal,  the  sides  opposite 
these  angles  are  equal,  and  the  triangle  is  isosceles. 

Corollary.     An  equiangular  triangle  is  also  equilateral. 

Proposition  XXXVI.    A  Theorem. 

73.  If  two  sides  of  a  triangle  are  unequal,  the  angle  op- 
posite the  longer  side  is  greater  than  the  angle  opposite  the 
shorter  side. 

Proposition  XXXVII.    A  Theorem.  , 

74.  If  two  angles  of  a  triangle  are  unequal,  the  side  op- 
posite the  greater  angle  is  longer  than  the  side  opposite  the 
lesser. 

Proposition  XXXVUI.    A  Theorem. 

75.  Two  right  triangles  are  equal  if  the  hypotenuse  and 
one  side  of  the  one  are  equal  to  the  hypotenuse  and  one 
side  of  the  other. 

Proposition  XXXIX.    A  Theorem. 

76.  Lines  bisecting  the  angles  of  a  triangle  meet  at  a  point 
which  is  equally  distant  from  the  side  of  the  triangle. 

See  Propositions  XXIII.  and  XXIV. 

Proposition  XL.    A  Theorem. 

77.  Perpendiculars  bisecting  the  sides  of  a  triangle  meet 
at  a  point  equally  distant  from  the  vertices  of  the  angles. 

See  Proposition  XVIII. 


1 6  PLANE   GEOMETRY. 


VII.   Quadrilaterals. 

78.   What  is  a  quadrilateral  or  quadrangle  ? 
a.    A  trapezium? 

Illustrations. 


b.  A  trapezoid  ? 

Illustrations.      /_ \^  \ >^ 

c.  A  parallelogram? 

d.  A  rectangle  ? 

e.  A  square? 
/.  A  rhombus  ? 


A. 


Illustration,     ^ 

g,  A  rhomboid  ? 

Illustration.      4 


^ 


8 

79.  What  is  the  diagonal  of  a  quadrilateral  ? 

80.  What  are  the  upper  and  lower  bases  of  a  quadrilateral  ? 

81.  What  is  the  altitude  of  a  parallelogram  or  trapezoid  ? 

82.  What  are  the  bases  of  a  trapezoid  ? 

a.    What  is  its  median  ? 

Illustration 


'A^^^- 


BOOK  I.  17 


Proposition  XLI.    A  Theorem. 

83.  The  opposite  sides  and  angles  of  a  parallelogram  are 
equal. 

Corollary  I.  The  diagonal  divides  a  parallelogram  into 
equal  triangles. 

Corollary  II.  The  parts  of  parallel  Unes  cut  off  between 
parallel  lines  are  equal. 

Proposition  XLII.    A  Theorem. 

84.  If  the  opposite  sides  of  a  quadrilateral  are  equal,  the 
figure  is  a  parallelogram. 

Proposition  XLIII.    A  Theorem. 

85.  If  two  sides  of  a  quadrilateral  are  equal  and  parallel, 
the  other  two  sides  are  also  equal  and  parallel,  and  the  figure 
is  a  parallelogram. 

Proposition  XLIV.    A  Theorem. 

86.  The  diagonals  of  a  parallelogram  bisect  each  other. 

Proposition  XLV.    A  Theorem. 

87.  Two  parallelograms  are  equal  if  they  have  two  sides 
and  the  included  angle  of  one  equal  to  two  sides  and  the 
included  angle  of  the  other,  each  to  each. 

Proposition  XLVI.    A  Theorem. 

88.  Parallel  lines  are  everywhere  equally  distant. 


1 8  PLANE   GEOMETRY. 


VIII.   Polygons  of  more  than  Four  Sides. 

89.  What  is  a  polygon  ? 

a.  A  pentagon  ? 

b.  A  hexagon? 

c.  A  heptagon? 

d.  An  octagon  ? 

e.  A  nonagon  ? 
/.  A  decagon  ? 

g.   An  undecagon  ? 
h.   A  duodecagon  ? 

90.  What  are  saHent  angles  of  a  polygon  ? 

91.  What  are  re-entrant  angles  ? 

92.  What  is  an  equilateral  polygon  ? 

93.  What  is  an  equiangular  polygon  ? 

94.  What  is  a  concave  polygon  ? 

95.  When  are  two  polygons  mutually  equiangular? 

96.  When  are  two  polygons  mutually  equilateral? 

97.  What  are  homologous  sides  or  angles  ? 

98.  What  are  equal  polygons  ? 

99.  When  is  a  polygon  symmetrical  with  reference  to  any 
dividing  line  ? 

100.  What  is  an  axis  of  symmetry  ? 

101.  What  is  a  centre  of  symmetry  ? 


BOOK  I.  19 


Proposition  XLVII.    A  Theorem. 

102.  Two  equal  polygons  may  be  divided  into  the  same 
number  of  equal  triangles. 

Proposition  XLVIil.    A  Theorem. 

103.  The  sum  of  the  interior  angles  of  a  polygon  is  equal 
to  as  many  right  angles  as  twice  a  number  two  less  than  the 
number  of  its  sides. 

Scholium.  To  how  many  right  angles  is  the  sum  of  the 
angles  of  figures  from  pentagons  to  duodecagons  equal  ?  If 
equiangular,  how  large  is  each  angle  ? 

Proposition  XLIX.    A  Theorem. 

104.  If  each  side  of  a  polygon  be  produced  in  order,  the 
sum  of  the  exterior  angles  equals  four  right  angles. 


OPTIONAL    PROPOSITIONS. 

Proposition  L.    A  Theorem. 

105.  I.  In  a  regular  polygon  having  an  odd  number  of 
sides,  a  line  joining  the  vertex  of  an  angle  with  the  middle 
point  of  the  opposite  side  is  an  axis  of  symmetry. 

II.  In  a  regular  polygon  having  an  even  number  of  sides, 
a  line  joining  the  vertices  of  opposite  angles,  or  the  middle 
points  of  opposite  sides,  is  an  axis  of  symmetry. 

Proposition  LI.    A  Problem. 

106.  Draw  parallel  lines  a  given  distance  apart. 


20  PLANE   GEOMETRY. 

Proposition  LI  I.    A  Theorem. 

107.  Any  number  of  parallel  lines  equally  distant  from 
each  other  intercept  equal  parts  on  any  transverse  line 
crossing  them. 

SUPPLEMENTARY   PROPOSITIONS. 

1.  What  is  the  supplement  to  an  angle  of  35°  ?  The 
complement  ? 

2.  If  three  or  more  angles  be  formed  at  the  same  point 
on  the  same  side  of  a  straight  line,  any  one  of  them  will  be 
a  supplement  to  the  sum  of  all  the  others. 

3.  If  two  adjacent  supplementary  angles  be  bisected,  the 
bisectors  will  form  a  right  angle. 

4.  A  line  bisecting  one  of  two  vertical  angles  will,  if  con- 
tinued, bisect  the  other. 

5.  The  sum  of  any  two  sides  of  a  triangle  is  greater  than 
the  sum  of  any  two  lines  drawn  from  any  point  in  the  tri- 
angle to  the  extremities  of  the  third  side. 

6.  An  equiangular  triangle  is  also  equilateral. 

7.  The  bisector  of  the  vertical  angle  of  an  isosceles  trian- 
gle, if  continued  to  the  base,  is  an  axis  of  symmetry. 

8.  In  a  right  triangle  the  two  acute  angles  are  comple- 
ments of  each  other. 

9.  In  a  right  triangle,  if  one  of  the  acute  angles  is  of  30°, 
the  side  opposite  is  one  half  the  hypotenuse. 

10.  Find  the  locus  of  a  point  equally  distant  from  two 
points. 

1 1 .  The  opposite  angles  of  a  parallelogram  are  equal. 


BOOK  I.  21 


12.  In  a  parallelogram,  the  angles  adjacent  to  any  one 
side  are  supplements. 

13.  All  the  angles  of  a  parallelogram  are  equal  to  four 
right  angles. 

14.  The  diagonals  of  a  rectangle  are  equal. 

15.  If  the  diagonals  of  a  quadrilateral  bisect  each  other, 
the  quadrilateral  is  a  parallelogram. 

16.  If  a  diagonal  divides  a  quadrilateral  into  two  equal 
triangles,  the  quadrilateral  is  a  parallelogram. 

17.  Through  a  given  point  to  draw  a  parallel  to  a  given 
line. 

18.  If  several  parallel  lines  intercept  equal  parts  on  any 
transverse  line,  they  are  an  equal  distance  apart. 

19.  If  a  line  drawn  through  a  triangle  parallel  to  one  of 
the  sides  bisects  one  of  the  other  sides,  it  will  bisect  both  of 
them. 

20.  A  line  bisecting  two  sides  of  a  triangle  is  parallel  to 
the  third. 

21.  A  line  connecting  the  middle  points  of  two  sides  of  a 
triangle  is  half  the  length  of  the  third. 

22.  The  medians  of  a  triangle  intersect  one  another  at 
the  same  point,  which  is  distant  from  the  vertex  of  each 
angle  two  thirds  the  length  of  its  median. 

23.  The  median  of  a  trapezoid  is  parallel  to  the  bases  and 
equally  distant  from  them. 

24.  The  median  of  a  trapezoid  is  equal  to  half  the  sum  of 
the  bases. 


BOOK   II. 

THE  PRINCIPLES  OF  PROPORTION  AND  THE 
THEORY  OF   LIMITS. 


I.  Ratio  and  Proportion. 

108.  What  is  a  ratio? 

a 

a.  How  expressed  ?     Example,    a  :  ^,  or  ^  • 

b.  What  names  are  given  the  terms  ? 

109.  What  is  a  proportion  ? 

a.  How  expressed?     Ex.    a-.bwcd,  or  j—  ■^' 

b.  What  names  are  given  the  terms  ? 

110.  What  is  a  fourth   proportional?     A  mean  propor- 
tional ?    A  third  proportional  ? 

111.  Given  a  proportion  a-.b-.-.c-.d,  what  is  changing  it : 

a.  By  inversion  ?     Ex.    b-.awd-.c. 

b.  By  alternation ?     Ex.    a:c\:b:  d,  or  d:b::c: a. 

c.  By  composition ?     Ex.   a  +  b-.b-.-.c  +  d-.d. 

d.  By  division?     Ex.    a  —  b-.bwc  —  d-.d. 

112.  What  common  divisor  has  8  and  34  ?    Their  ratio  ? 
What  common  divisor  has  3.6  and  54?     Their  ratio? 
What  common  divisor  has  5  and  ^40  ?    Their  ratio  ? 

113.  In  each  of  the  following  problems  how  long  a  line 
will  exactly  divide  both  the  given  lines? 

a.  Lines  10  and  25  inches,  respectively.  Their  ratio? 


BOOK  II.  23 


b.  Lines  8^  and  15  J  inches,  respectively.     Their  ratio  ? 

c.  Lines  6  and  Vs  inches,  respectively.     Their  ratio  ? 

114.  What  are  commensurable  quantities?  Incommen- 
surable quantities? 

115.  How  can  a  common  measure  and  the  ratio  of  two 
lines  be  found? 

116.  What  are  equimultiples  of  two  quantities? 

Proposition  I.     A  Theorem. 

117.  If  four  quantities  are  in  proportion,  the  product  of 
the  extremes  equals  the  product  of  the  means. 

Corollary.  A  mean  proportional  is  equal  to  the  square 
root  of  the  product  of  the  other  two  terms. 

Proposition  II.    A  Theorem. 

118.  If  two  sets  of  proportional  quantities  have  a  ratio  in 
each  equal,  the  other  ratios  will  be  in  proportion. 

Corollary.  If  the  antecedents  or  consequents  are  the 
same  in  both,  the  other  terms  are  in  proportion. 

Proposition  III.    A  Theorem. 

119.  If  the  product  of  two  quantities  equals  the  product  of 
two  other  quantities,  either  two  may  be  made  the  means  and 
the  other  two  the  extremes  of  a  proportion. 

Proposition  IV.    A  Theorem. 

120.  If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  inversion. 


24  PROPORTION. 


Proposition  V.    A  Theorem. 

121.   If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  alternation. 


Proposition  VI.    A  Theorem. 

122.   If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  composition. 


Proposition  VII.    A  Theorem. 

123.   If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  divdsion. 


Proposition  VIII.    A  Theorem. 

124.   If  four  quantities  are  in  proportion,  they  will  be  in 
proportion  by  composition  and  division. 


Proposition  IX.    A  Theorem. 

125.   Equimultiples  of  two  quantities  are  proportional  to 
the  quantities  themselves. 

Corollary.     Any  equimultiples  of  the   antecedents  are 
proportional  to  any  equimultiples  of  the  consequents. 


Proposition  X.    A  Theorem. 

126.   If  four  quantities  are  in  proportion,  their  like  powers 
or  like  roots  are  in  proportion. 


BOOK  II.  25 


Proposition  XI.    A  Theorem. 

127.  In  a  series  of  equal  ratios  the  sum  of  all  the  ante- 
cedents is  to  the  sum  of  all  the  consequents  as  any  one 
antecedent  is  to  its  consequent. 

Corollary.  The  sum  of  any  number  of  the  antecedents 
is  to  the  sum  of  their  consequents  as  any  one  antecedent  is 
to  its  consequent. 

Proposition  XII.    A  Theorem. 

128.  If  two  or  more  proportions  be  multiplied  together, 
term  by  term,  the  products  are  in  proportion. 


II.    The  Theory  of  Limits. 

129.  What  is  a  variable  ?     A  constant  ? 

a.  An  increasing  variable  ? 

b.  A  decreasing  variable  ? 

130.  Suppose  a  point  x  moving  on  the  line  A  B  m  such  a 

X  X  X      X         -, 

A — — — — — £> 

I  234 

way  that  it  goes  one  half  the  distance  from  A  to  B  the  first 
second,  one  half  the  remaining  distance  the  next  second, 
one  half  the  remaining  distance  the  third,  and  so  on  in- 
definitely : 

a.   What  two  varying  distances  does  it  produce  ? 

^.  What  distance  is  the  distance  A  x  approaching, 
and  when  will  it  reach  it  ? 

c.  What   is   the  distance  x  B  approaching,  and 

when  will  it  reach  it? 


26  THEORY  OF  LIMITS. 


13L    Reduce  the  fraction  ^  to  a  decimal : 

a.  How  is  the  value  of  the  decimal  affected  by 

each  division,  and  what  is  it  approaching? 

b.  How  is  the  difference  between  \  and  the  deci- 

mal affected  by  each  division,  and  what  is  it 
approaching? 

132.  What  is  the  limit  to  a  variable? 

a,  A  superior  limit? 

b.  An  inferior  limit  ? 

133.  How  near  may  a  variable  be  conceived  to  approach 
its  limit? 

134.  Suppose  a  right  triangle  to  be  continually  changing 
by  the  shortening  of  one  of  its  legs  : 

a.  What  lines  would  be  variables  ?     Their  limits  ? 

b.  What  angles  would  be  variables  ?    Their  limits  ? 

c.  How  would  its  area  be  affected  ?     Its  Hmit  ? 

135.  Why  could  not  the  diminishing  leg  become  zero  ? 

136.  Suppose  a  regular  polygon,  as  a  square  or  equilateral 
triangle,  to  be  inscribed  in  a  circle  (see  Book  III.  §  159),  and 
that  by  bisecting  the  arcs  and  drawing  chords  it  be  changed 
to  a  regular  inscribed  polygon  of  double  the  number  of  sides, 
four  times  the  number  of  sides,  and  so  on  indefinitely ; 

a.  What  variables  result  ? 

b.  What  are  their  limits  ? 

137.  Sometimes  the  variable  does  not  indefinitely  ap- 
proach limits,  as,  for  example,  suppose  the  process  in  §  136 
reversed. 


BOOK   II.  27 


Proposition  XIII.    A  Theorem. 

138.  If  two  variables  as  they  indefinitely  approach  their 
limits  have  any  constant  ratio,  their  limits  have  the  same 
ratio. 

Corollary.  If  two  variables  as  they  indefinitely  approach 
their  limits  are  constantly  equal,  their  limits  are  equal. 

Scholium.  In  the  above  corollary,  the  variables  have  the 
constant  ratio  i,  as  have  also  their  limits. 

Proposition  XIV.    A  Theorem. 

139.  If  the  product  of  two  variables  as  they  indefinitely 
approach  their  limits  is  constantly  equal  to  a  third  variable, 
the  products  of  their  limits  will  equal  the  limits  of  the  third. 

Scholium.     Sometimes  the  product  of  an  increasing  and  a 
decreasing  variable  is  a  constant. 
See  Book  IV.,  Proposition  XVII. 

Proposition  XV.    A  Theorem. 

140.  If  several  parallel  lines  are  crossed  by  an  oblique 
line,  the  segments  of  the  oblique  line  are  proportional  to  the 
distances  between  the  parallels. 

Case  I.    When  the  parallels  are  an  equal  distance  apart. 

Case  II.    When  the  parallels  are  unequal  distances  apart. 

a.  When  the  distances  between  them  are  com- 

mensurable. 

b.  When  these  distances  are  incommensurable. 

Corollary.  The  corresponding  segments  of  two  oblique 
transversals  are  in  proportion. 


28  PLANE   GEOMETRY. 

Proposition  XVI.    A  Theorem. 

141.  If  one  or  more  parallel  lines  be  drawn  through  a  tri- 
angle parallel  to  one  side,  the  other  two  sides  will  be  divided 
proportionally. 

Corollary.     The  intersected  sides  are  to  each  other  as 
any  two  corresponding  segments. 
See  Book  II,,  Proposition  VI. 

Proposition  XVII.    A  Theorem. 

142.  If  a  straight  line  divide  the  sides  of  a  triangle  propor- 
tionally, it  is  parallel  to  the  third  side. 

Proposition  XVIII.    A  Problem. 

143.  To  divide  a  given  line  into  parts  proportional  to 
given  lines,  or  given  parts  of  a  given  line. 

Proposition  XIX.    A  Problem. 

144.  To  find  a  fourth  proportional  to  three  given  lines. 

Proposition  XX.    A  Problem. 

145.  To  find  a  third  proportional  to  two  given  lines. 


BOOK    III. 


I.    The 

Circle. 

146. 

What  is  a  circle  ? 

147. 

What  is  the  circumference  ? 

148. 

What  is  the  radius? 

149. 

What  is  a  chord  ? 

Illustration. 

O 

150. 

What  is  a  diameter? 

151. 

What  is  a  secant? 

Illustration. 

o 

152. 

What  is  a  tangent  ? 

Illustration. 

<) 

153. 

What  is  an  arc  ? 

154. 

What  is  a  segment  ? 

Illustration. 

o 

155. 

What  is  a  sector? 

Illustration. 

(!) 

a.   A  quadrant? 

Illustration. 

(3 

156.   When  do  circles  touch  each  other  internally?    When 
externally  ? 


30  PLANE   GEOMETRY. 

157.  When  is  an  angle  inscribed  in  a  circle?       in.  (/\ 

158.  When  is  an  angle  inscribed  in  a  segment  ?    m. 


159.  When  is  a  polygon  inscribed  in  a  circle  ?    m. 

a.  What  is  the  relation  of  the  circle  to  the  polygon } 

160.  When  is  a  circle  inscribed  in  a  polygon  ?    in. 

a.  What  is  the  relation  of  the  polygon  to  the  circle  ? 

161.  What  are  concentric  circles?      Illustration. 

162.  When  will  circles  be  equal? 

163.  What  is  an  angle  at  the  centre  ? 

Proposition  I.    A  Theorem. 

164.  The  diameter  of  a  circle  is  an  axis  of  symmetry. 

Corollary.     The  diameter  bisects  the  circle  and  its  cir- 
cumference. 

Proposition  li.    A  Theorem. 

165.  A  straight  line  cannot  intersect  the  circumference  of 
a  circle  at  more  than  two  points. 

See  Book  I.,  Proposition  XVI.,  Corollary. 

Proposition  III.    A  Theorem. 

166.  The  diameter  is  longer  than  any  other  chord. 


BOOK  III.  31 


Proposition  IV.    A  Theorem. 

167.  In  the  same  circle,  or  in  equal  circles,  equal  arcs  are 
subtended  by  equal  chords ;  and  conversely,  if  the  chords 
are  equal,  the  arcs  also  are  equal. 

Proposition  V.    A  Theorem. 

168.  In  the  same  circle,  or  in  equal  circles,  of  two  une- 
qual arcs  each  less  than  a  semicircumference,  the  greater 
arc  is  subtended  by  the  longer  chord ;  and  conversely,  of 
two  unequal  chords,  the  longer  subtends  the  greater  arc. 

See  Book  I.,  Propositions  XXXI.  and  XXXII. 

Proposition  VI.    A  Theorem. 

169.  A  radius  drawn  perpendicular  to  a  chord  bisects  both 
the  chord  and  its  arc. 

Scholium.  A  line  drawn  perpendicular  to  the  middle 
point  of  a  chord  is  a  radius  and  bisects  the  arc. 


Proposition  VII.    A  Theorem. 

L  three  given  points  not  in  1 
line  one  circumference  can  be  drawn,  and  but  one 


170.   Through  three  given  points  not  in  the  same  straight 


Proposition  VIII.     Problems. 

171.    I.  Given  three  points  not  in  a  straight  line,  to  draw 
a  circumference  through  them. 

See  Book  I.,  Propositions  XVIII.  and  XIX. 
II.    Given  a  circumference,  to  find  its  centre. 


32  PLANE   GEOMETRY. 

Proposition  IX.    A  Theorem. 

172.  In  the  same  circle,  or  in  equal  circles,  equal  chords 
are  equally  distant  from  the  centre ;  and  conversely,  chords 
equally  distant  from  the  centre  are  equal. 

See  Book  I.,  Proposition  XXXVIII. 

Proposition  X.     A  Theorem. 

173.  In  the  same  circle,  or  in  equal  circles,  of  two  unequal 
chords  the  shorter  is  the  farther  from  the  centre. 

See  Book  I.,  Proposition  XIV. 

Proposition  XI.    A  Theorem. 

174.  Conversely,  of  two  chords  unequally  distant  from  the 
centre,  the  farther  one  will  be  the  shorter. 

Proposition  XII.    A  Theorem. 

175.  A  tangent  is  perpendicular  to  the  radius  drawn  to 
the  point  of  tangency. 

See  Book  I.,  Proposition  XIV. 

Corollary.  Conversely,  a  straight  line  perpendicular  to  a 
radius  at  its  termination  in  the  circumference  is  a  tangent  to 
the  circle. 

Proposition  XIII.    A  Theorem. 

176.  If  from  a  point  outside  of  a  circle  two  tangents  to 
the  circle  be  drawn,  and  also  a  straight  line  to  the  centre  of 
the  circle  : 

I.   The  tangents  will  be  equal. 

II.  The  line  drawn  to  the  centre  bisects  the  angle  formed 
by  the  tangents. 

See  Book  I.,  Proposition  XXIV. 


BOOK   III.  33 


Proposition  XIV.    A  Theorem. 

177.  Parallels  intercept  on  a  circumference  equal  arcs. 
Case  I.     When  both  parallels  are  tangents. 

Case  II.     When  one  is  a  tangent  and  the  other  a  secant. 
Case  III.     When  both  are  secants. 

Proposition  XV.    A  Theorem. 

178.  I.  If  two  circles  cut  each  other,  the  line  joining  their 
centres  will  be  perpendicular  to  their  common  chord. 

II.  If  two  circles  touch  each  other  externally  or  internally, 
the  line  joining  their  centres  will  be  perpendicular  to  their 
common  tangent 

Proposition  XVI.    A  Theorem. 

179.  In  the  same  circle,  or  in  equal  circles,  radii  forming 
equal  angles  at  the  centre  intercept  equal  arcs  on  the  circum- 
ference ;  and  conversely,  if  the  arcs  intercepted  are  equal,  the 
angles  at  the  centre  are  equal. 

Proposition  XVII.    A  Theorem. 

180.  In  the  same  circle,  or  in  equal  circles,  angles  at  the 
centre  are  to  each  other  as  their  arcs. 

Case  I.     When  they  are  commensurable. 

Case  II.     When  they  are  incommensurable. 

Scholium  I.   The  angle  may  be  measured  by  the  arc ;  why? 

Scholium  II.  Explain  the  division  of  the  arc  into  de- 
grees, etc. 

Scholium  III.  What  arc  measures  a  right  angle?  An 
acute  angle?     An  obtuse  angle? 

3 


34  PLANE   GEOMETRY. 

Proposition  XVIII.    A  Theorem. 

181.  An  inscribed  angle  is  measured  by  half  the  inter- 
cepted arc. 

Case  I.  When  one  of  the  chords  forming  the  angle  is  the 
diameter. 

See  Book  I.,  Proposition  X. 

Case  II.  When  the  chords  are  on  opposite  sides  of  the 
centre. 

Case  III.  When  both  chords  are  on  the  same  side  of  the 
centre. 

Corollary  I.  All  angles  inscribed  in  the  same  segment 
are  equal. 

Corollary  II.     All  angles  inscribed  in  a  semicircle  are 

right  angles. 

Corollary  III.  An  angle  inscribed  in  a  segment  smaller 
than  a  semicircle  is  obtuse;  one  inscribed  in  a  segment 
greater  than  a  semicircle  is  acute. 

Proposition  XIX.    A  Theorem. 

182.  The  angle  formed  by  two  chords  which  intersect  each 
other  is  measured  by  half  the  sum  of  the  included  arcs. 

Proposition  XX.    A  Theorem. 

183.  The  angle  formed  by  two  secants  is  measured  by 
half  the  difference  of  the  included  arcs. 

Proposition  XXI.    A  Theorem. 

184.  The  angle  formed  by  a  tangent  and  a  chord  is  meas- 
ured by  half  the  intercepted  arc. 


BOOK  III. 


35 


Proposition  XXII.    A  Theorem. 

185.  An  angle  formed  by  two  tangents  is  measured  by 
half  the  difference  of  the  intercepted  arcs. 

Note.  In  circles  that  are  not  equal,  radii  forming  equal  angles  at 
the  centre  intercept  arcs  whose  absolute  length  is  not  the  same,  but 
they  contain  the  same  number  of  degrees,  and  may  be  called  homolo- 
gous. This  can  be  easily  shown  by  means  of  concentric  circles. 
Hence,  in  any  circles,  (a)  radii  forming  equal  angles  at  the  centre, 
(d)  equal  inscribed  angles,  (c)  equal  angles  formed  by  intersecting 
chords,  by  secants,  or  by  tangents,  intercept  homologous  arcs ;  and, 
conversely,  if  the  arcs  intercepted  are  homologous,  the  angles  are 
equal.  It  is  on  this  principle  that  Supplementary  Propositions  13, 
14,  and  15  of  this  Book  depend. 

Proposition  XXIII.    A  Problem 

186.  To  erect  a  perpendicular  at  the  end  of  a  given  line. 
See  Book  III.,  Proposition  XVIII.,  Corollary  II. 

Proposition  XXIV.    A  Problem. 

187.  At  a  point  on  a  line,  to  construct  an  angle  equal  t» 
a  given  angle. 

See  Book  III.,  Proposition  XVII.,  Scholium  I.,  and  Propo- 
sition IV. 

Proposition  XXV.    A  Problem. 

188.  To  bisect  a  given  arc. 
See  Book  III.,  Proposition  VI. 


Proposition  XXVI.    A  Problem. 
189.    To  bisect  a  given  angle. 


36  PLANE   GEOMETRY. 

Proposition  XXVII.    A  Problem. 

'  190.   Through  a  given  point  to  draw  a  line  parallel  to  a 
given  line. 

See  Book  I.,  Proposition  XI. 

Proposition  XXVIII.    A  Problem. 

191.  Two  angles  of  a  triangle  being  given,  to  find  the 
third. 

Proposition  XXIX.    A  Problem. 

192.  To  construct  a  triangle  when  two  of  its  sides  and  the 
angle  included  by  them  are  given. 

Proposition  XXX.    A  Problem. 

193.  Given  a  side  and  two  angles,  to  construct  the  triangle. 

Proposition  XXXI.    A  Problem. 

194.  Given  two  sides  of  a  triangle  and  the  angle  opposite 
one  of  them,  to  construct  the  triangle. 

Proposition  XXXII.    A  Problem. 

195.  Given  the  three  sides,  to  construct  the  triangle. 
See  Book  I.,  Proposition  XXX. 

Proposition  XXXIII.    A  Problem. 

196.  To  construct  a  parallelogram  when  its  adjacent  sides 
and  their  included  angle  are  given. 


BOOK   III.  37 


Proposition  XXXIV.    A  Problem. 

197.  From  a  given  point  to  draw  a  tangent  to  a  given 
circle. 

See  Book  III.,  Proposition  XII.,  and  Proposition  XVIII., 
Corollary  II. 

Proposition  XXXV.    A  Problem. 

198.  At  a  given  point  in  the  circumference  of  a  circle  to 
draw  a  tangent. 

See  Book  III.,  Proposition  XXIII. 

Proposition  XXXVI.    A  Problem. 

199.  In  a  given  triangle  to  inscribe  a  circle. 
See  Book  I.,  Proposition  XXXIX. 

Proposition  XXXVII.    A  Problem. 

200.  The  chord  being  given,  to  construct  a  circle  such 
that  any  angle  inscribed  in  one  of  the  segments  will  be  equal 
to  a  given  angle. 

See  Book  III.,  Proposition  XXI.,  Proposition  XVIII.,  Co- 
rollary I.,  and  Proposition  XII. 

Proposition  XXXVIII.    A  Problem. 

201.  Two  arcs  or  two  angles  being  given,  to  find  their 
common  measure. 

Proposition  XXXIX.     A  Theorem.     {Optional.) 

202.  The  side  of  an  inscribed  equilateral  triangle  and  the 
radius  perpendicular  to  it  bis'ect  each  other.  ' 


38  PLANE   GEOMETRY. 


SUPPLEMENTARY   PROPOSITIONS. 

1.  From  any  point  in  a  circle  the  shortest  distance  to  the 
circumference  will  be  on  the  radius  passing  through  the 
point. 

2.  From  any  point  in  a  circle  the  farthest  distance  to 
the  circumference  will  be  on  the  line  passing  through  the 
centre. 

3.  If  a  circle  is  touched  internally  by  another  circle  having 
half  the  diameter,  any  chord  of  the  larger  circle  drawn  from 
the  point  of  contact  is  bisected  by  the  circumference  of  the 
smaller  circle. 

4.  The  shortest  chord  that  can  be  drawn  through  any 
point  in  a  circle  is  the  one  drawn  at  right  angles  to  the 
radius  passing  through  the  point. 

5.  The  opposite  angles  of  an  inscribed  quadrilateral  are 
supplements  of  each  other. 

6.  If  the  opposite  angles  of  a  quadrilateral  are  supple- 
ments of  each  other,  a  circumference  can  be  circumscribed 
about  it. 

7.  The  sides  of  an  inscribed  equilateral  triangle  are  half 
the  length  of  the  sides  of  a  similar  circumscribed  triangle. 

See  Book  I.,  Supplementary  Propositions  19,  20,  and  21. 

8.  If  two  circles  intersect  each  other,  the  distance  be- 
tween their  centres  is  less  than  the  sum  and  greater  than 
the  difference  of  their  radii. 

9.  The  sum  of  the  opposite  arcs  intercepted  by  two 
chords  crossing  each  other  at  right  angles  equals  a  semi- 
circumference. 


BOOK  III.  39 


10.  If  two  equal  circles  intersect  each  other,  parallel 
secants  passing  through  the  points  of  intersection  cut  off 
reciprocally  equal  arcs  and  segments. 

11.  If  two  equal  intersecting  circles  are  cut  by  two  se- 
cants passing  through  the  points  of  intersection,  chords 
subtending  the  exterior  arcs  intercepted  by  these  secants 
will  be  parallel. 

Case  I.     When  the  secants  do  not  cross. 

Case  II.  When  the  secants  cross  each  other  in  one  of 
the  circles. 

12.  If  two  equal  circles  touch  each  other,  two  secants 
passing  through  the  point  of  contact,  will  intercept  equal 
arcs ;  and  the  chords  subtending  these  arcs  will  be  parallel. 

13.  If  two  unequal  circles  intersect  each  other,  two 
parallels  passing  through  the  points  of  intersection  and  ter- 
minated by  the  exterior  arcs,  will  be  equal. 

See  Note,  page  35. 

14.  If  two  unequal  intersecting  circles  are  cut  by  secants 
passing  through  the  points  of  intersection,  chords  subtending 
the  exterior  arcs  intercepted  are  parallel. 

Case  I.     When  the  secants  do  not  cross. 

Case  II.  When  the  secants  cross  each  other  in  one  of  the 
circles. 

See  Note,  page  35. 

15.  If  two  unequal  circles  touch  each  other,  two  secants 
passing  through  the  point  of  contact  will  intercept  homolo- 
gous arcs,  and  the  chords  subtending  these  arcs  will  be 
parallel. 


40  PLANE   GEOMETRY. 


1 6.  Construct  an  angle  of  60°  ;  of  120°  ;  of  30°  ;  of  15°. 

17.  Construct  an  angle  of  45°.     Divide  it  into  three  equal 
angles. 

18.  Divide  a  right  angle  into  three  equal  angles. 

19.  Find  a  point  equidistant  from  three  given  points. 

20.  Find  a  point  equidistant  from  two  given  points,  and  a 
given  distance  from  a  third  given  point. 

21.  Construct  a  perpendicular  from  the  vertex  of  one 
angle  of  a  triangle  to  the  opposite  side. 

22.  Divide  a  line  into  three  equal  parts. 
See  Book  I.,  Supplementary  Proposition  18. 

23.  Given  the  radius  and  two  points  in  the  circumference, 
to  construct  the  circle. 

24.  A  chord  and  a  point  in  the  circumference  given,  to 
construct  the  circle. 

25.  To  lay  off  on  a  given  circumference  an  arc  of  180°  ; 
of  90°;  of  60°;  of  30°;  of  120°. 

26.  The  base,  the  altitude,  and  one  of  the  angles  at  the 
base  given,  to  construct  the  triangle. 

27.  Given  one  side,  the  diagonal,  and  the  included  angle, 
to  construct  a  parallelogram. 

28.  In  a  given  circle  to  inscribe  an  equilateral  triangle. 

29.  About  a  given  circle  to  circumscribe  an  equilateral 
triangle. 

30.  The  radius  is  two  thirds  the  altitude  of  an  inscribed, 
and  one  third  the  altitude  of  a  circumscribed  equilateral 
triangle. 


BOOK  III.  41 


31.  Find  in  a  given  circumference  two  points  such  that 
tangents  passing  through  them  will  meet  at  an  angle  of  30°. 

32.  Find  in  a  given  circumference  two  points  such  that 
two  tangents  passing  through  them  will  meet  at  an  angle 
of  90°. 

33.  Given  the  perimeter  and  altitude  of  a  triangle,  and 
the  point  on  the  perimeter  where  the  perpendicular  from 
the  opposite  angle,  which  equals  the  altitude,  would  fall,  to 
construct  the  triangle. 

34.  To  construct  a  triangle,  the  base,  altitude,  and  angle 
at  the  vertex  being  given. 

See  Book  III.,  Proposition  XXXVII. 

35.  To  construct  a  triangle,  the  base,  angle  at  the  vertex, 
and  median  connecting  them  being  given. 

36.  From  a  given  point  draw  tangents  to  a  given  circle ; 
connect  these  tangents  by  a  line  drawn  tangent  to  the  smaller 
intercepted  arc  ;  a  triangle  will  be  formed,  the  sum  of  whose 
sides  will  be  constant  at  whatever  point  on  the  arc  the  con- 
necting tangent  be  drawn. 

See  Book  III.,  Proposition  XIII. 

37.  If,  with  the  conditions  as  given  in  36,  lines  be  drawn 
from  the  centre  of  the  circle  to  the  extremities  of  the  con- 
necting tangent,  the  angle  at  the  centre  will  remain  constant 
through  all  positions  of  the  tangent. 

38.  To  construct  a  right  triangle,  when  given  : 

a.   Hypotenuse  and  one  side. 

h.    Hypotenuse  and  altitude  on  the  hypotenuse. 

c.   One  side  and  altitude  on  the  hypotenuse. 


42  PLANE   GEOMETRY. 

39.  To  construct  a  scalene  triangle,  when  given  : 

a.  The  perimeter  and  angles. 

b.  One  side,  an  adjacent  angle,  and  sum  of  the 

other  sides. 

c.  The  sum  of  two  sides  and  the  angles. 

d.  The  angles  and  the  radius  of  an  inscribed  circle. 

e.  An  angle,  its  bisector,  and  the  altitude  from  the 

given  angle. 

40.  To  construct  a  rectangle,  when  given  : 

a.  One  side  and  the  angle  formed  by  the  diagonals. 

b.  The  perimeter  and  a  diagonal. 

41.  To  construct  a  rhombus,  when  given  : 

a.  One  side  and  the  radius  of  the  inscribed  circle. 

b.  One  angle  and  the  radius  of  the  inscribed  circle. 

42.  To  construct  a  rhomboid,  when  given  : 

a.  One  side  and  the  two  diagonals. 

b.  The  base,  the  altitude,  and  one  angle. 

43.  To  construct  a  trapezoid,  when  given  : 

a.  The  bases,  the  altitude,  and  one  angle. 

b.  One  base,  the  adjacent  angles,  and  one  side. 

c.  One  base,  the  adjacent  angles,  and  the  median. 


BOOK    IV. 


I.   Similar  Polygons. 

203.  When  are  polygons  similar?  * 

204.  What  are  their  homologous  parts  ? 

205.  What  is  meant  by  their  ratio  of  simiHtude  ? 

Proposition  I.     A  Theorem. 

206.  Two  mutually  equiangular  triangles  are  similar. 
See  Book  II.,  Proposition  XVI. 

Corollary.  Triangles  having  two  angles  mutually  equal, 
or  an  angle  in  each  equal  and  the  sides  including  it  in  pro- 
portion, are  similar. 

Proposition  II.    A  Theorem. 

207.  If  triangles  have  their  sides  taken  in  order  in  propor- 
tion, they  are  similar. 

Proposition  III.    A  Problem. 

208.  The  ratio  of  the  homologous  sides  being  equal  to  the 
ratio  of  two  given  Hnes,  to  construct  a  triangle  similar  to  a 
given  triangle. 

*  Form  what  proportions  you  can  from  two  similar  triangles  ;  from 
two  similar  quadrilaterals ;  from  two  similar  pentagons. 


44  PLANE   GEOMETRY. 


Proposition  IV.    A  Theorem. 

209.  Two  triangles  whose  sides  are'parallel  or  perpendicu- 
lar are  similar. 

Proposition  V.    A  Theorem. 

210.  Two  similar  polygons  may  be  divided  into  the  same 
number  of  triangles,  similar  each  to  each. 

Proposition  VI.    A  Theorem. 

211.  If  two  polygons  can  be  divided  into  the  same  num- 
ber of  triangles,  similar  each  to  each,  and  similarly  placed, 
the  two  polygons  are  similar. 

Proposition  VI  i.    A  Problem. 

212.  A  polygon  being  given,  on  a  line  corresponding  to 
one  of  its  sides,  to  construct  a  similar  polygon. 

Proposition  VIII.    A  Theorem. 

213.  The  perimeters  of  two  similar  polygons  have  the  same 
ratio  as  any  two  homologous  sides. 

Proposition  IX.    A  Theorem. 

214.  Any  number  of  straight  Hues  intersecting  at  a  com- 
mon point  intercept  proportional  segments  on  two  parallels. 

Case  I.  When  the  parallels  are  on  the  same  side  of  the 
common  point. 

Case  II.     When  they  are  on  opposite  sides. 

Note.  This  principle  may  be  used  in  finding  the  sides  in  Propo- 
sition VII. 


BOOK   IV.  45 


Proposition  X.    A  Theorem. 

215.  Conversely,  all  non-parallel  lines  intercepting  pro- 
portional segments  on  two  parallel  lines  intersect  at  a 
common  point. 


II.  Division  of  Lines. 

216.   What  is  dividing  a  line  internally  ? , 
Example,    a  — — 


a.  What  are  the  segments? 

217.   What  is  dividing  a  line  externally? 
Example.       


a.   What  are  the  segments  ? 

SPECIAL  PROBLEMS. 

218.  a.   To  divide  a  line  internally  in  the  ratio  of  2:3; 

of  3  ;  5  ;  of  2  :  7  j  etc. 
See  Proposition  IX.,  or  Book  II.,  Proposition  XVI 1 1. 

^.   To  divide  a  line  externally  in  the  ratio  of  2  :  3 ; 
of  3  :  5  ;  of  2  :  7  ;  etc. 

219.  What  is  dividing  a  line  harmonically? 

Example. 

c 

Divided  externally  as 

SPECIAL  PROBLEMS. 

220.  To  divide  a  given  line  harmonically  in  the  ratio  of 
3:4;  of  3  :  5  ;  of  2  :  7  ;  etc. 


Divided  internally  as 

A 

1^ 

to  s 

1  r  1  1 

'                         ' 

to  5 

1 



46  PLANE   GEOMETRY. 

Proposition  XI.    A  Theorem. 

221.  A  line  bisecting  an  angle  of  a  triangle  divides  the 
opposite  side  into  segments  proportional  to  the  adjacent 
sides  including  the  angle. 

Proposition  XII.    A  Theorem. 

222.  A  line  bisecting  an  exterior  angle  of  a  triangle  divides 
the  opposite  side -externally  into  segments  proportional  to  the 
other  two  sides. 

Proposition  XIII.    A  Theorem. 

223.  Lines  bisecting  adjacent  interior  and  exterior  angles 
of  a  triangle  divide  the  opposite  side  harmonically. 

Proposition  XIV.    A  Problem. 

224.  To  divide  a  line  harmonically. 

Proposition  XV.    A  Theorem. 

225.  In  a  right  triangle,  if  a  perpendicular  is  drawn  from 
the  vertex  of  the  right  angle  to  the  hypotenuse : 

I.   The  right  triangle  is  divided  into  two  triangles 
similar  to  itself  and  to  each  other. 
II.   The  perpendicular  is  a  mean  proportional  be- 
tween the  segments  of  the  hypotenuse. 
III.    Each  side  of  the  right  angle  is  a  mean  propor- 
tional between  the  whole  hypotenuse  and  the 
adjacent  segment. 
Corollary  I.    The  squares  of  the  sides  of  the  right  angle 
are  proportional  to  the  adjacent  segments  of  the  hypotenuse. 

Corollary  II.  The  square  of  the  hypotenuse  of  a  right 
triangle  is  equivalent  to  the  sum  of  the  squares  of  the  other 
two  sides. 


BOOK  IV.  47 


Proposition  XVI.    A  Problem. 

226.  To  find  a  mean  proportional  between  two  given 
lines. 

Proposition  XVII.    A  Theorem. 

227.  The  products  of  the  two  segments  of  all  chords 
drawn  through  any  fixed  point  in  a  circle  are  constant.* 

Proposition  XVIII.    A  Theorem. 

228.  From  a  point  without  a  circle,  in  whatever  direction 
a  secant  is  drawn,  the  product  of  the  whole  secant  by  its 
external  segment  is  constant. 

Proposition  XIX.    A  Theorem. 

229.  If  from  a  point  without  a  circle  a  secant  and  a  tan- 
gent be  drawn,  the  tangent  is  a  mean  proportional  between 
the  whole  secant  and  its  external  segment. 

How  can  this  be  proved  by  the  Theory  of  Limits  ? 


III.  Extreme  and  Mean  Ratio. 

230.    What  is  dividing  a  line  in  extreme  and  mean  ratio  ? 

Examples. 

C 

Internally.        A B 


AB  :AC..AC'.  CB. 

Externally.  A. 


CB'.CA'.'.CA  '.AB. 


*  Find  a  line  proportional  to  three  given  lines  by  this  principle. 


48  PLANE   GEOMETRY. 

Proposition  XX.    A  Problem. 

231.  To  divide  a  given  line  in  extreme  and  mean  ratio. 
See  Proposition  XIX.,  and  Book  II.,  Proposition  VII. 

Proposition  XXI.    A  Theorem. 

232.  In  any  triangle,  the  product  of  any  two  sides  is  equal 
to  the  product  of  the  perpendicular  to  the  third  side  from  the 
opposite  angle  by  the  diameter  of  the  circumscribed  circle. 

Proposition  XXII.    A  Theorem. 

233.  If  an  angle  of  a  triangle  be  bisected  by  a  line  termi- 
nating in  the  opposite  side,  the  product  of  the  segments  of 
this  side  plus  the  square  of  the  bisector  equals  the  product  of 
the  other  two  sides. 

Proposition  XXIII.    A  Theorem. 

234.  Homologous  altitudes  of  similar  triangles  are  propor- 
tional to  any  two  homologous  sides. 

SUPPLEMENTARY   PROPOSITIONS. 

1.  The  chord 
A  B  bisects  the 
common  tangent 
CD. 

2.  The  com- 
mon tangent  CD 
is  a  mean  pro- 
portional between 
the  diameters  of 
the  circles. 


BOOK   IV.  49 


3.  If  two  circles  intersect  each  other,  the  common  chord 
produced  bisects  the  common  tangent. 

4.  If  the  common  chord  of  two  intersecting  circles  be 
produced,  tangents  drawn  from  any  point  in  it  to  the  circles 
are  equal. 

5.  To  inscribe  a  square  in  a  given  triangle. 

6.  To  inscribe  a  square  in  a  semicircle. 

7.  To  inscribe  in  a  given  triangle  a  rectangle  similar  to  a 
given  rectangle. 

8.  To  circumscribe  about  a  circle  a  triangle  similar  to  a 
given  triangle. 

9.  To  construct  a  circle  whose  circumference  will  be  tan- 
gent to  a  given  line  and  pass  through  two  given  points. 

10.  To  construct  a  circle  whose  circumference  will  be  tan- 
gent to  two  given  lines  and  pass  through  one  given  point. 


BOOK    V. 

MEASUREMENT  AND   COMPARISON   OF 
RECTILINEAR   FIGURES. 


I.  Area. 

235.  What  is  area? 

a.    How  measured? 

Proposition  I.     A  Theorem. 

236.  The  area  of  a  rectangle  is  equal  to  the  product  of  its 
base  and  altitude. 

Case  I.     When  the  base  and  altitude  are  commensurable. 

Case  II.     When  they  are  incommensurable. 

Proposition  II.    A  Theorem. 

237.  The  area  of  a  parallelogram  is  equal  to  the  product 
of  its  base  and  altitude. 

Corollary.     Parallelograms  having  equal  bases  and  alti- 
tudes are  equivalent. 


BOOK  V.  51 


Proposition  III.    A  Theorem. 

238.  Parallelograms  are  to  each  other  as  the  products  of 
their  respective  bases  and  altitudes. 

Corollary.  Parallelograms  having  equal  altitudes  are  to 
each  other  as  their  bases  j  those  having  equal  bases  are  to 
each  other  as  their  altitudes. 

Proposition  IV.    A  Theorem. 

239.  The  area  of  a  triangle  is  equal  to  half  the  product  of 
its  base  and  altitude. 

Corollary.  Triangles  having  the  same  base  and  altitude 
are  equivalent 

Proposition  V.    A  Theorem. 

240.  Triangles  are  to  each  other  as  the  products  of  their 
respective  bases  and  altitudes. 

Corollary.  Triangles  having  the  same  altitudes  are  to 
each  other  as  their  bases,  and  those  having  the  same  bases 
are  to  each  other  as  their  altitudes. 

Proposition  VI.    A  Theorem. 

241.  The  area  of  a  trapezoid  is  equal  to  the  product  of  its 
altitude  by  half  the  sum  of  its  bases,  or  by  its  median. 

Proposition  VII.    A  Theorem. 

242.  The  areas  of  two  triangles  having  an  angle  in  each 
equal  are  to  each  other  as  the  products  of  the  sides  including 

n    the  equal  angle. 


52  PLANE  GEOMETRY. 

Proposition  VIII.    A  Theorem. 

243.  The  square  described  on  the  sum  of  two  lines  is 
equal  to  the  sum  of  their  squares  plus  two  rectangles  con- 
tained by  the  lines. 

Scholium.     Compare  {a  +  dy  :=  a^  +  2  al?  +  d\ 

Proposition  IX.    A  Theorem. 

244.  The  square  described  on  the  difference  of  two  lines 
is  equal  to  the  sum  of  their  squares  minus  two  rectangles 
contained  by  the  hnes. 

Scholium.     Compare  (a  —  dy  =  a^  —  2  ad  +  P. 

Proposition  X.    A  Theorem. 

245.  The  rectangle  contained  by  the  sum  and  difference 
of  two  lines  is  equal  to  the  difference  of  their  squares. 

Scholium.     Compare  {a  +  d)  {a  —  b)  =  a^  —  b"^. 

Proposition  XI.    A  Theorem. 

246.  The  square  described  on  the  hypotenuse  of  a  right 
triangle  is  equal  to  the  sum  of  the  squares  of  the  other 
two  sides. 

Corollary.  The  square  described  on  either  side  forming 
the  right  angle  is  equal  to  the  square  of  the  hypotenuse 
minus  the  square  of  the  other  side. 

Proposition  XII.    A  Problem. 

247.  To  construct  a  square  equal  to  the  sum  of  two  given 
squares. 


BOOK  V.  53 


Proposition  XI 11.    A  Problem. 

248.  To  construct  a  square  equal  to   the  difference  of 
two  given  squares. 

Proposition  XIV.    A  Problem. 

249.  To  construct  a  square  equal  to  the  sum  of  any  given 
number  of  given  squares. 

Proposition  XV.    A  Problem. 

250.  I.  To   construct  a  square    equivalent  to  a  given 
rectangle. 

See  Book  IV.,  Proposition  XVI. 


II.   Equivalent  to  a  given  triangle. 


Proposition  XVI.    A  Problem. 

251.   The  sum  of  the  base  and  altitude  given,  to  construct 
a  parallelogram  equivalent  to  a  given  square. 


Proposition  XVII.    A  Problem. 

252.   The  difference  between  the  base  and  altitude  given, 
to  construct  a  parallelogram  equivalent  to  a  given  square. 


Proposition  XVIII.    A  Theorem. 

253.    The  areas  of  similar  triangles  are  to  each  other  as 
the  squares  of  their  homologous  sides. 
See  Proposition  VII. 


54  PLANE  GEOMETRY. 


Proposition  XIX.    A  Theorem. 

254.   The  areas  of  any  similar  polygons  are  proportional  to 
the  squares  of  their  homologous  sides. 


Proposition  XX.    A  Problem. 

255.  I.  To  construct  a  polygon  similar  to  two  given  poly- 
gons but  equal  to  their  sum. 

See  Proposition  XI L 

II.    Equal  to  their  difference. 

Proposition  XXI.    A  Problem. 

256.  I.  To   construct  a  triangle  equivalent  to  a  given 
polygon. 

II.   To  construct  a  square  equivalent  to  a  given  polygon 
of  five  or  more  sides. 

Proposition  XXII.    A  Problem. 

257.  To  construct  a  square  having  a  given  ratio  to  a 
given  square. 

Proposition  XXIII.    A  Problem. 

258.  In  a  given  ratio  between  their  areas,  to  construct  a 
polygon  similar  to  a  given  polygon. 

Proposition  XXIV.    A  Problem. 

259.  To  construct  a  polygon  similar  to  one  given  polygon 
but  equivalent  to  another. 

See  Proposition  XXI. 


BOOK    V. 


55 


II.  Projection. 

260.   What  is  the  projection  of  a  point  on  a  line  ? 
Illustrations. 


The  point.  A 


The  point. 


261.   What  is  the  projection  of  a  line  on  another  line  ? 
Illustrations. 


B 


Proposition  XXV.    A  Theorem. 

262.  In  a  triangle,  the  square  of  a  side  opposite  an  acute 
angle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides  minus  twice  the  rectangle  formed  by  one  of  these  sides 
and  the  projection  of  the  other  on  it. 


Proposition  XXVI.    A  Theorem. 

263.  In  a  triangle,  the  square  of  the  side  opposite  an  ob- 
tuse angle  is  equal  to  the  sum  of  the  squares  of  the  other 
two  sides  plus  twice  the  rectangle  formed  by  one  of  these 
sides  and  the  projection  of  the  other  on  it. 


56  PLANE   GEOMETRY. 


Proposition  XXVII.    A  Theorem. 

264.  In  a  triangle,  if  a  median  line  is  drawn  from  the  ver- 
tex of  any  angle ; 

I.  The  sum  of  the  squares  of  the  sides  including  the  angle 
is  equal  to  twice  the  square  of  the  median  plus  twice  the 
square  of  half  the  side  it  bisects. 

II.  The  difference  of  the  squares  of  the  two  sides  includ- 
ing the  angle  is  equal  to  twice  the  rectangle  formed  by  the 
third  side  and  the  projection  of  the  median  on  it. 

Corollary  I.  In  any  quadrilateral  (not  a  parallelogram) 
the  sum  of  the  squares  of  the  four  gides  is  equal  to  the  sum 
of  the  squares  of  the  diagonals  plus  four  times  the  square  of 
the  line  joining  the  middle  points  of  the  diagonals. 

Corollary  II.  In  a  parallelogram  the  sum  of  the  squares 
of  the  four  sides  is  equal  to  the  sum  of  the  squares  of  the 
diagonals. 

SPECIAL   PROBLEMS. 

1.  Express  the  altitude  of  an  equilateral  triangle  in  terms 
of  its  sides. 

Suggestion.    Let  a  be  the  length  of  one  side,  and  x  the  altitude. 

2.  Express  the  area  of  an  equilateral  triangle  in  terms  of 
its  sides. 

Suggestion.    Let  A  represent  the  area. 

3.  Express  the  altitude  of  any  triangle  in  terms  of  its  sides. 

Suggestion.  Let  a,  b,  and  c  represent  the  lengths  of  the  different 
sides. 


BOOK  V.  57 


4.  Express  the  area  of  a  triangle  in  terms  of  its  sides. 

5 .  Express  a  median  of  a  triangle  in  terms  of  its  sides. 

6.  Express  the  bisector  of  an  angle  of  a  triangle  in  terms 
of  its  sides. 

7.  Express  the  radius  of  a  circle  circumscribed  about  a 
triangle  in  terms  of  the  sides  of  the  triangle. 


BOOK   VI. 


I.  Regular  Polygons. 

265.  What  is  a  regular  polygon? 

a.  The  apothem? 

Proposition  I.    A  Theorem. 

266.  A  circle  may  be  circumscribed  about  and  one  In- 
scribed within  a  regular  polygon. 

Corollary.    The  radius  drawn  to  the  vertex  of  an  angle 
of  a  regular  inscribed  polygon  bisects  the  angle. 

Proposition  II.     A  Theorem. 

267.  An  equilateral  polygon  inscribed  in  a  circle  is  regular. 

Proposition  III.    A  Theorem. 

268.  If  a  circumference  be  divided  into  equal  arcs  : 

I. .  The    chords    subtending    these   arcs   form   a  regular 
polygon. 

IL  The  tangents  drawn  at  the  points  of  division  form  a 
regular  polygon. 

See  Book  III.,  Proposition  XI IL 


BOOK  VI.  59 


Proposition  IV.    A  Problem. 

269.  I.   Given  a  regular  inscribed  polygon,  to  construct 
one  having  double  the  number  of  sides. 

II.    Given  a  regular  circumscribed  polygon,  to  construct 
one  having  double  the  number  of  sides. 

Proposition  V.    A  Problem. 

270.  In  a  given  circle  to  construct  a  square. 


Proposition  VI.    A  Theorem. 

271.   The  side  of  a  regular  inscribed  hexagon  is  equal  to 
the  radius  of  the  circumscribed  circle. 


Proposition  VII.     A  Problem. 

272.  To  inscribe  a  regular  hexagon  in  a  given  circle. 

Proposition  VIII.    A  Problem. 

273.  To  inscribe  a  regular  decagon  in  a  given  circle. 
See  Book  IV.,  Proposition  XX. 

Scholium.     Inscribe  a  regular  pentagon. 

Proposition  IX.    A  Problem. 

274.  To  inscribe  a  regular  pentedecagon  in  a  given  circle. 

Proposition  X.    A  Theorem. 

275.  Two  regular  polygons  of  the  same  number  of  sides 
are  similar. 


6o  PLANE   GEOMETRY. 

Proposition  XI.    A  Theorem. 

276.  The  perimeters  of  two  regular  polygons  of  the  same 
number  of  sides  are  to  each  other  : 

I.    As  their  sides. 

II.    As  the  radii  of  circumscribed  circles. 

III.   As  the  radii  of  inscribed  circles. 

Corollary.  If  in  two  circles  all  possible  regular  polygons 
be  drawn,  the  perimeters  of  those  in  one  circle  will  have  to 
the  perimeters  of  the  similar  ones  in  the  other  circle  a  con- 
stant ratio. 

Proposition  XII.     A  Theorem. 

277.  The  circumferences  of  circles  are  to  each  other  as 
their  radii  or  their  diameters. 

6"/?^  Book  II.,  §§  120-128. 

Corollary.  The  ratio  of  circumferences  to  their  radii  or 
to  their  diameters  is  constant. 

Scholium  k  The  constant  ratio  of  the  circumference  to 
the  diameter  is  represented  by  the  Greek  letter  n,  and  it  will 
be  hereafter  one  of  our  objects  to  ascertain  its  numerical 
value. 

Scholium  II.  Let  2  J?  represent  the  diameter ;  the  cir- 
cumference will  be  2  17  R. 

Proposition  XIII.    A  Theorem. 

278.  The  areas  of  two  regular  polygons  of  the  same  num- 
ber of  sides  are  to  each  other  : 

I.   As  the  squares  of  their  sides. 

II.   As  the  squares  of  the  radii  of  circumscribed  circles. 

III.    As  the  squares  of  the  radii  of  inscribed  circles. 


BOOK  VI.  6l 


Proposition  XIV.    A  Theorem. 

279.  The  areas  of  circles  are  to  each  other  as  the  squares 
of  their  radii  or  of  their  diameters. 

Corollary.  The  areas  of  similar  sectors  or  segments  are 
to  each  other  as  the  squares  of  the  radii  or  of  the  diameters. 

Proposition  XV.    A  Theorem. 

280.  I.  The  difference  between  the  perimeters  of  regular 
inscribed  and  circumscribed  polygons  of  the  same  number  of 
sides  is  indefinitely  diminished  as  the  sides  are  indefinitely 
multiplied. 

II.  The  difference  between  their  areas  is  indefinitely 
diminished  as  the  sides  are  indefinitely  multiplied. 

Proposition  XVI.    A  Theorem. 

281.  The  area  of  a  regular  polygon  is  equal  to  half  the 
product  of  its  perimeter  by  its  apothem. 

Proposition  XVII.    A  Theorem. 

282.  The  area  of  a  circle  is  equal  to  half  the  product  of 
the  circumference  by  the  radius. 

Scholium.  U  2  h  R  (see  Proposition  XII.,  Scholium  II.) 
represents  the  circumference,  what  will  be  the  area  of  the 
circle  ? 

Corollary.  The  area  of  a  sector  is  equal  to  half  the 
product  of  its  arc  and  the  radius. 


62  PLANE  GEOMETRY. 


Proposition  XVIII.      A  Problem. 

283.  Given  the  radius  and  a  chord,  to  compute  the  chord 
of  half  the  arc  subtended. 

Scholium.  This  principle  can  be  used,  when  the  side  of  a 
regular  inscribed  polygon  is  known,  to  find  the  side,  and 
therefore  the  perimeter,  of  a  regular  polygon  of  double  the 
number  of  sides. 

Proposition  XIX.     Problems. 

284.  I.  To  find  the  ratio  between  the  perimeter  of  a  regu- 
lar inscribed  hexagon  and  the  diameter  of  the  circle. 

II.  Between  the  perimeter  of  a  regular  inscribed  duodeca- 
gon  and  the  diameter  of  the  circumscribed  circle. 


Proposition  XX.    A  Problem. 

285.  To  compute  the  numerical  value  of  n. 

OPTIONAL  PROPOSITIONS. 

Proposition  XX 1.     A  Problem. 

286.  The  perimeters  of  a  regular  inscribed  and  a  similar 
circumscribed  polygon  being  known,  to  compute  the  perime- 
ters of  the  regular  inscribed  and  circumscribed  polygons  of 
double  the  number  of  sides. 

Proposition  XXII.     A  Problem. 

287.  To  compute  the  numerical  value  of  ir  from  the  pre- 
ceding problem. 


BOOK  VI.  63 


SPECIAL   PROBLEMS. 
Proposition  XXIII.    A  Problem. 

288.  L    Express  the  side  of  an  inscribed  equilateral  tri- 
angle in  terms  of  the  radius. 

IL   The  same  of  a  regular  inscribed  hexagon. 
IIL   The  same  of  a  regular  inscribed  duodecagon. 
Note.     Continue  this  as  far  as  desirable. 

Scholium.     What  would  the  perimeters  be  in  each  case  ? 

Corollary.     Express  the  areas  of  the  above  in  terms  of 
the  radius. 

Proposition  XXIV.    Problems. 

289.  L    Express  the  side  of  an  inscribed  square  in  terms 
of  the  radius. 

IL   The  same  of  a  regular  inscribed  octagon. 
Note.     Continue  as  far  as  desirable. 

Scholium.     What  would  the  perimeters  be  in  each  case  ? 
Corollary.     Express  their  areas  in  terms  of  the  radius. 

Proposition  XXV.     Problems. 

290.  I.   Express   the  side  (and   perimeter)  of  a  regular 
inscribed  decagon  in  terms  of  the  radius. 

See  Proposition  VIII. 

II.   Express  the  side  of  a  regular  inscribed  pentagon  in 
terms  of  the  radius. 

Corollary.     Express  their  areas  in  terms  of  the  radius. 

Proposition  XXVI.    A  Problem. 

291.  Compute  the  numerical  value  of  tt  from  one  of  the 
above  problems. 


64  PLANE    GEOMETRY. 

SUPPLEMENTARY   PROBLEMS. 
Geometrical  Construction  of  Algebraic  Equations. 

Note.  In  these  problems,  the  first  letters  of  the  alphabet  express 
known,  or  given  lines  ;  in  performing  operations  with  them,  the  fol- 
lowing points  should  be  kept  in  mind : 

1.  The  product  of  two  lines,  or  the  square  of  a  line  is  a  surface. 

2.  The  product  of  three  lines  is  a  solid. 

3.  A  surface  divided  by  a  line,  is  a  line. 

4.  The  square  root  of  the  product  of  two  lines  is  a  line. 

The  letter  x  represents  the  element  to  be  constructed 
and  may  be  a  line,  surface,  or  solid  as  the  case  requires. 
Problem  L     Construct  x  =  a  ■\-  d. 
Problem  II.     Construct  x  —  a  —  b. 
Problem  III.     Construct  x  —  ab. 
Problem  IV.     Construct  xz=abc. 

Problem  V.     Construct  ^  =  —  . 

c 

See  Book  IL,  Proposition  XIX. 

Problem  VI.     Construct  x  —  -r. 

o 

See  Book  IL,  Proposition  XX. 


Problem  VII.     Construct  x  =  ^ a^  ■\-  b'^. 
See  Book  V.,  Proposition  XL 


Problem  VIII.     Construct  x  =  ^  a^  —  b^. 
Problem  IX.     Construct  x  =  ^~ab. 

See  Book  IV.,  Proposition  XVI. 

Problem  X.     Construct  x  =  ^  d^  —  a  b. 
See  Book  IV.,  Proposition  XIX. 


SUPPLEMENTARY    PROBLEMS.  65 


Problem  XI.     Construct  x  —  a  ±  's/  a^  —  b^. 

SuG.  Construct  a  line  equal  to  a,  and  at  one  extremity 
construct  a  perpendicular  equal  to  b.  With  the  remote  end 
of  <^  as  a  centre  and  a  radius  equal  to  a,  draw  an  arc  cutting 
a  and  a  prolonged. 

Problem  XII.     Form  the  equation  for  the   larger  seg- 
ment of  a  line  a  divided  in  extreme  and  mean  ratio. 
See  Book  IV.,  Proposition  XX. 

Problem  XIII.  Form  the  equation  for  the  side  of  a 
square  inscribed  in  a  triangle  whose  base  and  altitude  are 
given. 

Problem  XIV.  Given  the  radii  and  the  distance  be- 
tween the  centres  of  two  unequal  circles,  form  the  equation 
for  the  distance  to  the  point  where  their  common  tangents 
will  meet. 


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